How do I solve for x squared plus y squared minus six x minus ten y plus 9 equals 0?

That is the formula for a circle: complete the squares in x and y to find the centre and the radius:

A circle with centre (x0, y0) and radius r has the formula:

(x - x0)² + (y - y0)² = r²

Completing the square is a technique to convert x² - ax into a perfect square less a constant c, ie:

x² - ax = (x - b)² - c

Now:

(x - b)² = x² - 2bx + b²

→ x² - 2bx = (x - b)² - b²

If b = a/2, this becomes:

x² - ax = (x - a/2)² - (a/2)²

So back to the given equation:

x² + y² - 6x - 10y + 9 = 0

→ (x² - 6x) + (y² - 10y) + 9 = 0

→ ((x -3)² - 3²) + ((y - 5)² - 5²) + 9 = 0

→ (x - 3)² + (y - 5)² + (-9 - 5² + 9) = 0

→ (x - 3)² + (y - 5)² - 5² = 0

→ (x - 3)² + (y - 5)² = 5²

→ It is a circle with centre (3, 5) and radius 5.